Trusses are pin-jointed structures made of 2-force members (where each member is pure tension or compression).A truss becomes a frame if it contains a member with 3 or more forces (a beam). These members can ber in bending - and the force is no longer aligned through the 2 joints of a member. Most frame problems are a mixture of 2-force (tension/compression) and multi-force members.

Notes 2011:   Frames.pdf

Notes 2010:   Frames1.pdf

AutoCad Sample: Frames-001.dwg


Amazingly, every pin-jointed structure can be solved with the same old technique we learnt in Non-Concurrent Forces:

  • "Apply equilibrium to any portion of the structure".
What does "apply equilibrium" mean? Simply this...

What does "any portion of the structure" mean?

Exactly what it says! You can choose any "portion" you like - such as the whole structure, half the structure, a group of members or one single member.

For example. We can apply equilibrium to the whole structure, like this...

(Yes, I know you've seen it before, but it is so important that you need to see it again!)


(From Ivanoff, Ch8. Pin Reactions in Frames)
  1. Considering the frame as a whole, determine all support reactions in terms of their horizontal and vertical components.
  2. Identify any two-force members in the frame. It is important to recognise these as having two equal and opposite forces applied through the pins and acting along the axis of such members. This helps to recognise the lines of action of forces acting on other connected members.
  3. Isolate each separate member of the frame and draw it as a free body, showing all known and unknown forces acting on the member. As usual, the sense of an unknown force may be assumed, and then revised in the light of subsequent calculations.
  4. Select a member, with at least one known force, which contains no more than three unknowns. use the three equations of statics to calculate the unknown pin-reaction forces acting on the member.
  5. With the knowledge gained from the previous step, select another member and solve for more unknowns. It is essential to remember that a reaction force acting on any given member at a joint is the pulling or pushing action from another member, transmitted through the joint. Between any two members connected through a pin, action and reaction forces are always equal and opposite.
  6. Repeat step 5 until all components of pin-reaction forces have been determined.
  7. As a final step, if desired, combine horizontal and vertical components of pin—reaction forces at each joint into their resultant in order to detemine the total magnitude of the reaction force at the joint. ‘

In other words...

Solve equilibrium for each member treated as a beam (non-concurrent forces), and wherever you find a 2-force member the direction of that force is axial (lines up through it's two joints).
You have to find a member (beam) that can be solved (can take moments about some point that leaves only 1 unknown).
Once this is done, move on to the next member etc.

In other other words...

We can solve just about anything in Statics by;

1. Pick out a section as a FBD

2. Take moment equation at well-chosen location that leaves one unknown.

3. Solve other unknowns for that FBD using force polygon (Which is ΣFx=0 & ΣFy=0)


Easy Huh?

Hint: The Three Force Principle comes in mighty handy here. Or you may prefer to use X and Y components of forces at every joint throughout the truss. Which is easier? Hmmm - about the same either way.

Example. (See example 8.1, page 122 of text; Ivanoff Engineering Mechanics)


Super Summary of 2D Statics

  • Take an appropriate Free Body Diagram, then solve equilibrium. Continue until finished.
  • Solving equilibrium depends on the number of unknowns - which can be a Force Magnitude, Force Direction or a Moment.

Two ways to solve ANY Free Body Diagram headache: (Do not induce vomiting).


Force Polygon (2 unknowns)
Moment Equation (3 unknowns)

Solve Mathematically;

Or solve graphically (CAD);

Moment Equation

+ (10*4) + (10*8) - (Fg*24) = 0


Common Free Body Diagrams


Type Point Member with 2 forces Whole Structure Half Structure Member with 3+ forces
Solvable if... 2 unknowns or less 1 unknown (trivial) 3 unknowns or less 3 unknowns or less 3 unknowns or less
Looks like...


F1 + F2 = 0
Solve by...

Force Polygon

Take Opposite

F1 = -F2

Moment Equation

then F.P.

Moment Equation

then F.P.

Moment Equation

then F.P.


Number of Unknowns for Various Joints


Lesson Whiteboards... (based on Ivanoff example 8.1)

And now you are FINISHED!!! (Well, once you have passed the tests of course)
Speaking of tests, here's a whiteboard full of hints for the Container Crane questions:


Homework Assignment:
  • Do all questions 8.1 to 8.4 (page 125-126).
Exam Rules:
Permitted: Open Book, Internet, Calculator, CAD
Not Permitted: Excel, any dedicated truss analysis software, pre-programmed solutions - including VisualBasic, Excel etc.
Relevant pages in MDME (Pre-requisites)
  • Engineering Mechanics: Equilibrium By C. Hartsuijker, J. W. Welleman. (Chapter 9 Trusses)